Wave Coherence and Interference

For stable interference patterns, the two sources must be coherent, meaning they must have the same frequency and a constant phase relationship.

When two waves are propagate in same direction

Phase difference is $\phi (t)$,

Since, the phase difference
is varying, then the waves are said to be incoherent. So, the intensity of
resultant wave is the sum of intensities of each wave i.e. $2,{{I} _{0}}$ 

Given

Constructive interference for coherent waves takes place when phase difference = $2n\pi$ where n is a integer

In a light source, a light wave(photon) is produced when an excited atom goes to the ground state and emits light.

In a light source(photon) is produced when an excited atom goes to the ground state and emits light.

Two coherent sources are produced in Lloyd's mirror by reflection of light from a plane mirror, hence the original source and its image acting as coherent sources.

The atoms in the laser medium are pumped up energetically by an external power source, producing a population inversion of atomic energies. The pumped-up atoms can return to lower energy levels by emitting light at only certain frequencies, depending on the particular medium, which can be a solid (e.g. laser diode), a liquid (e.g. dye laser) or a gas (e.g. helium-neon laser).

Two source are called coherent, if they have constant phase difference i.e., phase difference between two wave is constant with respect to time.

Interference is a phenomenon in which two waves superpose to form a resultant wave of greater or lower amplitude.

All of the above type of waves.

Two sources are said to be coherent if they have exactly same frequency and have zero or constant phase difference. Two coherent source cannot be obtained from two different light sources.

Coherent sources are typically derived from a single point source (e.g., using a beam splitter or double slits) to ensure they share the same initial phase and frequency.

Since, interference occurs both in sound and light waves so it will occur in all the forms of waves given.

Laser sources produce light waves which are coherent. As sustained interference takes place between waves, which are coherent, we can use two independent laser sources for the same.

LASER refers to light amplification based on the stimulated emission of Radiation. It emits light coherently. Thus the photons emitted in a laser beam are coherent.

When the light waves are emitted from a single source and they have a constant phase difference between them then the waves are said to be coherent.
Option C is correct .

Interference refers to the interaction of waves that are correlated of coherent with each other. Sustained interference takes place between waves which are coherent.

Interference of light from two sources can be observed if the sources are coherent . It means that the phase difference between two sources must be constant with time . Two independent sources can't produce an interference pattern .

Laser light is considered to be coherent because it consist of waves of exactly the same wavelength in phase.

To obtain a well defined interference patterns, the intensity at points corresponding to destructive interference must be zero, while intensity at the point corresponding to constructive interference must be maximum. To accomplish this the necessary condition is that the two interference sources must be coherent, that is, they must keep a constant phase difference.

For observing interference the term identical source means that their wavelength are the same ( i.e., they are coherent).

By reflection we create a virtual source from real source and light from both of these sources are used to create interference pattern in Lloyd's single mirror method.

For sources to be coherent it is necessary that their initial phase difference remains the same or constant.

the coherence can be  obtained by using sources 3 and 4 as only the intensities are different so their frequencies can be equal
for a coherent sources frequencies must be equal
option $B$ is correct 

All points on the wavefront have the same temporary displacement. Then need not be oscillating with the same frequency and constant phase difference.
Two wave sources are said to be coherent if they produce waves of same frequency and constant phase difference.
Hence assertion is incorrect, reason is correct. Option A.

For sustainable interference wavelength has to be same. So frequency has to be same too. Contact phase relation is also necessary to sustain the interference pattern else the pattern will keep changing so rapidly that we will not see any pattern.

There will be general illumination as super imposing waves do not have constant phase difference.

When the red and green filters are used, the rays from slits will have wavelengths of red and green light.  The condition of mono chromatic sources will not be met. Hence, the fringe pattern will disappear.

In physics, two wave sources are perfectly coherent if they have a constant phase difference and the same frequency (amplitude may be different).

For interference to take place source must be coherent, which implies:

• Same frequency for the two waves
• Constant phase difference.

The number of wavelengths the wave travels is $=\dfrac{l}{\lambda}$ $=\dfrac{2\times 10^{-6}}{400\times 10^{-10}}$ $=50$     which is an integer.

For demonstration of interference of sound we need two coherent  source, of same frequency with constant phase difference

$\displaystyle \frac{I _2}{I _1}=\beta$                        $\displaystyle \frac{I _{min}}{I _{max}}=\frac{\beta - 2 \sqrt{\beta}+1}{\beta + 2 \sqrt{\beta +1}}$
$\frac{A _2}{A _1} =\sqrt{\beta}$
$\displaystyle \frac{A _2 -A _1}{A _2 + A _1}=\frac{\sqrt{\beta}-1}{\sqrt{\beta}+1}$     $\displaystyle \frac{I _{max}- I _{min}}{I _{max}+ I _{min}}=\frac{(2)}{(2)} \left [ \frac{2 \sqrt{\beta}}{\beta +1}\right ]$

Coherent sources are created by splitting light from a single source, ensuring that any phase changes in the original source are reflected equally in both secondary sources, maintaining a constant phase difference.

$n\lambda =\left( \mu -1 \right) t$\ $20\times 5\times  { 10 }^{ -7 }=\left( \mu -1 \right) 25\times { 10 }^{ -6 }$\ $\mu =1.4$